L(s) = 1 | − 1.40·2-s − 17.9·3-s − 30.0·4-s + 25·5-s + 25.1·6-s + 73.4·7-s + 87.1·8-s + 78.5·9-s − 35.1·10-s + 121·11-s + 538.·12-s − 1.19e3·13-s − 103.·14-s − 448.·15-s + 838.·16-s + 1.12e3·17-s − 110.·18-s + 361·19-s − 750.·20-s − 1.31e3·21-s − 169.·22-s + 140.·23-s − 1.56e3·24-s + 625·25-s + 1.67e3·26-s + 2.94e3·27-s − 2.20e3·28-s + ⋯ |
L(s) = 1 | − 0.248·2-s − 1.15·3-s − 0.938·4-s + 0.447·5-s + 0.285·6-s + 0.566·7-s + 0.481·8-s + 0.323·9-s − 0.111·10-s + 0.301·11-s + 1.07·12-s − 1.95·13-s − 0.140·14-s − 0.514·15-s + 0.818·16-s + 0.940·17-s − 0.0802·18-s + 0.229·19-s − 0.419·20-s − 0.651·21-s − 0.0748·22-s + 0.0553·23-s − 0.553·24-s + 0.200·25-s + 0.485·26-s + 0.778·27-s − 0.531·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.7822088456\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7822088456\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
good | 2 | \( 1 + 1.40T + 32T^{2} \) |
| 3 | \( 1 + 17.9T + 243T^{2} \) |
| 7 | \( 1 - 73.4T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.19e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.12e3T + 1.41e6T^{2} \) |
| 23 | \( 1 - 140.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.57e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.62e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 3.75e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 644.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.21e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.07e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.77e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 700.T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.17e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.34e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.53e5T + 8.58e9T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.424427804079988342579111810255, −8.356046445818953256434904770752, −7.56265554146243945455731835639, −6.60061865585849619739384799062, −5.45373863092685076563935120586, −5.10107947470865168802957566342, −4.30020265507646421489734248367, −2.80941757265867611180581812784, −1.39796494249050255222961220397, −0.46328814248716978618535389126,
0.46328814248716978618535389126, 1.39796494249050255222961220397, 2.80941757265867611180581812784, 4.30020265507646421489734248367, 5.10107947470865168802957566342, 5.45373863092685076563935120586, 6.60061865585849619739384799062, 7.56265554146243945455731835639, 8.356046445818953256434904770752, 9.424427804079988342579111810255